Domains whose ideals meet a universal restriction

Abstract

Let D be an integral domain, S(D) = I(D) (It(D)) the set of proper nonzero ideals (proper t-ideals) of D, M ax(D) (t-M ax(D) the set of maximal (t-) ideals of D, and let P be a predicate on S(D) with nonempty truth set Π S(D) ⊆ S(D), where P can be: "-is invertible" or "-is divisorial" etc.. We say S(D) meets P (S(D)We show that if S(D) ⊳ P, we have no control over dim D. We also show that I(D) ⊳ P does not imply I(R) ⊳ P, while It(D) ⊳ P implies It(R) ⊳ P, for most choices of P, when R = D[X] and have examples to show that generally S(D) ⊳ P does not extend to rings of fractions. We study restrictions that may control the dimension of D when S(D) ⊳ P. We also say S(D) ⊳ P with a twist (S(D) ⊳ t P ) if ∀s ∈ S(D) ∃π ∈ Π D (P )(s n ⊆ π for some n ∈ N ) and study S(D) ⊳ t P, along the same lines as S(D) ⊳ P and provide examples.